How to Make Implicit Upwind Schemes as Efficient as Streamline … · 2014. 11. 17. · How to Make Implicit Upwind Schemes as Efﬁcient as Streamline Methods J. R. Natvig†, K.-A

How to Make Implicit Upwind Schemes asEfficient as Streamline Methods

J. R. Natvig†, K.-A. Lie† B. Eikemo‡

†SINTEF, Department of Applied Mathematics

‡Department of Mathematics, University of Bergen

CMWR XVI, Copenhagen, June 20 2006

CMWR, Copenhagen 1/ 22

Topic of this talk

Aim:To construct fast implicit solver for

φ∂S

∂t+ v · ∇f(S) = Q(S), (1)

assuming no gravity and no capillary pressure.

Method:Decompose (1) in sequence of local problems.


Implicit First-Order Upwind Scheme 1D

S1 S2v

Q1 Q2

φ

∆t(Sn

1 −Sn−11 )− v

∆xf(Sn

1 ) = Q1(Sn1 ),

φ

∆t(Sn

2 −Sn−12 ) +

v

∆xf(Sn

1 ) = Q2(Sn2 ),



S1 S2v

Q1 Q2

φ

∆t(Sn

1 −Sn−11 )− v

∆xf(Sn

1 ) = Q1(Sn1 ),

φ

∆t(Sn

2 −Sn−12 ) +

v

∆xf(Sn

1 ) = Q2(Sn2 ),



S1 · · · Sk−1 Sk Sk+1 · · · Sm

v1 vk−1 vk vm−1

Consider the scheme

φ

∆t(Sn

k−Sn−1k ) +

1

∆x

(vk−1f(Sn

k−1)− vkf(Snk )

)= Qk(Sn

k ),

where we have assumed vk > 0.



S1 · · · Sk−1 Sk Sk+1 · · · Sm

v1 vk−1 vk vm−1

Consider the scheme

φ

∆t(Sn − Sn−1) +

1

∆x

−v1

v1 −v2

. . . . . .vm−1 0

f(Sn

1 )...

f(Snm)

= Q(S).




S1 · · · Sk−1 Sk Sk+1 · · · Sm

v1 vk−1 vk vm−1

Consider the scheme

φ

∆t(Sn − Sn−1) +

1

∆x

−v1

v1 −v2

. . . . . .vm−1 0

f(Sn

1 )...

f(Snm)

= Q(S).


Lower triangular matrix =⇒ equations can be solved in sequence.



Then consider the same scheme in 2D withm grid cells and fluxes given by the (sparse)m×m-matrix v.

k jvkj

Snk − Sn−1

k

∆t− 1

h

( ∑j

max(vkj , 0)f(Snk ) +

∑j

min(vkj , 0)f(Snj )

)= Qk(Sn

k ).

Again, this can be written in matrix notation

1

∆t(Sn − Sn−1) + V

f(Sn1 )

...f(Sn

m)

= Q(Sn).

where

Vkk = − 1

h

∑j

max(vkj , 0), Vkj = − 1

hmin(vkj , 0).

Is V triangular?




k jvkj


1

∆t(Sn − Sn−1) + V

f(Sn1 )

...f(Sn

m)

= Q(Sn).

where

Vkk = − 1

h

∑j

max(vkj , 0), Vkj = − 1

hmin(vkj , 0).

Is V triangular?




k jvkj


1

∆t(Sn − Sn−1) + V

f(Sn1 )

...f(Sn

m)

= Q(Sn).

where

Vkk = − 1

h

∑j

max(vkj , 0), Vkj = − 1

hmin(vkj , 0).

Is V triangular?


“Homogeneous” Quarter five-spot

What does V look like?

1 2 3

4 5 6

7 8 9

V =

∗∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗∗ ∗ ∗

∗ ∗∗ ∗ ∗

∗ ∗ ∗

Lower triangular


“Homogeneous” Quarter five-spot

What does V look like?

1 2 3

4 5 6

7 8 9

V =

∗∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗∗ ∗ ∗

∗ ∗∗ ∗ ∗

∗ ∗ ∗

Lower triangular


“Heterogeneous” Quarter five-spot

What does V look like now?

1 2 3

4 5 6

7 8 9

V =

∗∗ ∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗∗ ∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗

Not lower triangular


“Heterogeneous” Quarter five-spot

What does V look like now?

1 2 3

4 5 6

7 8 9

V =

∗∗ ∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗∗ ∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗

Not lower triangular


Graph Interpretation

What can be done?


Graph Interpretation Continued

“Homogeneous” Quarter five-spot as directed graph

1 2 3

4 5 6

7 8 9

Can this directed graph be flattenedsuch that all arrows point to the right?

1 2 3 4 5 6 7 8 9

Yes!




1 2 3

4 5 6

7 8 9


1 2 3 4 5 6 7 8 9

Yes!




1 2 3

4 5 6

7 8 9


1 2 3 4 5 6 7 8 9

Yes!



“Heterogeneous” Quarter five-spot as directed graph

1 2 3

4 5 6

7 8 9

What about this directed graph?




1 2 3

4 5 6

7 8 9

What about this directed graph?

1 2 3 4 5 6 7 8 9


Topological sorting

Topological sorting:

Find sequence of vertex numbers (p1, . . . , pm) such that

pi < pj

whenever there is a directed edge from vertex i to vertex j.

A topological sort of the vertices in a directed graph can be found inlinear time as the post-order of the depth-first traversal of thereversed graph.




1 2 3

4 5 6

7 8 999

87

4

1

4

7 8

6

32

5

2 3

6

9

1 4 7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 9988

7

4

1

4

7 8

6

32

5

2 3

6

9

1 4 7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 99877

4

1

4

7 8

6

32

5

2 3

6

9

1 4 7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 9987

44

1

4

7 8

6

32

5

2 3

6

9

1 4 7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 9987

4

1

1

4

7 8

6

32

5

2 3

6

9

1 4 7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 9987

4

1

4

4

7 8

6

32

5

2 3

6

9

1

4 7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 998

7

4

1

4

7

7 8

6

32

5

2 3

6

9

1 4

7 8 5 2 3 6 9




1 2 3

4 5 6

7 8 998

7

4

1

4

7 8

8

6

32

5

2 3

6

9

1 4 7

8 5 2 3 6 9




1 2 3

4 5 6

7 8 998

7

4

1

4

7 8 9

6

32

5

2 3

6

9

1 4 7 8

5 2 3 6 9




1 2 3

4 5 6

7 8 998

7

4

1

4

7 8

66

32

5

2 3

6

9

1 4 7 8

5 2 3 6 9




1 2 3

4 5 6

7 8 998

7

4

1

4

7 8

6

33

2

5

2 3

6

9

1 4 7 8

5 2 3 6 9




1 2 3

4 5 6

7 8 99

87

4

1

4

7 8

6

322

5

2 3

6

9

1 4 7 8

5 2 3 6 9




1 2 3

4 5 6

7 8 99

87

4

1

4

7 8

6

32

5

5

2 3

6

9

1 4 7 8

5 2 3 6 9




1 2 3

4 5 6

7 8 99

87

4

1

4

7 8

6

32

5

2

2 3

6

9

1 4 7 8 5

2 3 6 9




1 2 3

4 5 6

7 8 99

87

4

1

4

7 8

6

3

2

5

2 3

3

6

9

1 4 7 8 5 2

3 6 9




1 2 3

4 5 6

7 8 99

87

4

1

4

7 8

6

32

5

2 3

6

6

9

1 4 7 8 5 2 3

6 9




1 2 3

4 5 6

7 8 9

987

4

1

4

7 8

6

32

5

2 3

6

9

9

1 4 7 8 5 2 3 6

9




1 2 3

4 5 6

7 8 9

987

4

1

4

7 8

6

32

5

2 3

6

9

1 4 7 8 5 2 3 6 9


Triangularisation of the coefficient matrix

By rearranging the rows and columns in the order(1, 4, 7, 8, 5, 2, 3, 6, 9), we obtain a triangular V :

1 2 3

4 5 6

7 8 9

1 4 7 8 5 2 3 6 9147852369

∗∗ ∗

∗ ∗∗ ∗

∗ ∗ ∗∗ ∗ ∗

∗ ∗∗ ∗ ∗

∗ ∗ ∗

Duff and Reid. An implementation of Tarjans algorithm for block triangularisation of a matrix. 1978.

Dennis, Martinez and Zhang. Triangular decomposition methods for solving reducible nonlinear systems. 1994.


Triangularisation of the coefficient matrix


1

∆t(S̃n − S̃n−1) + L

f(S̃n1 )

...f(S̃n

m)

= PQ(S̃n).

where S̃ = PS, L = PV P T and P is a permutation matrixobtained from a topological ordering of the grid cells.



k jvkj

Snk − Sn−1

k

∆t− 1

h

( ∑j

max(vkj , 0)f(Snk ) +

∑j

min(vkj , 0)f(Snj )

)= Qk(Sn

k ).



k jvkj

Find uh ∈ Vh such that∫K

Snk − Sn−1

k

∆tvh−

∫K

f(Snk ) v·∇vh+

∑j

∫∂K

vhf̂(Snk , Sn

j , vkj)=

∫K

Qk(Snk )vh,

For all vh ∈ Vh.

Here f̂ is the upwind flux given by

f̂(Sk, Sj , vkj) = f(Sk) max(vkj , 0) + f(Sj) min(vkj , 0).


Delineation of Reservoirs

Consider a domain with multiple injectors at positions (x1, . . . ,xn).Solve

v · ∇Ci = Qi

where Qi > 0 for the injector at injector xi and zero elsewhere.

Ci(x) =

{1 x on streamline from injector i,

0 otherwise.


Porosity of the model 2 of the SPE Comparative Solution Project.


Delineation of Reservoirs ContinuedModel 2 of the SPE Comparative Solution Project*.

Size: 60× 220× 85 (1.122 mill. grid blocks)Simulation time: a few minutes.

Christie and Blunt Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques


Twophase Flow in 2D

First-order upwind 2nd-order discontinuous Galerkin

Solution after 0.2 PVI .

Solution after 0.3 PVI.

Water flooding in layer 6 of the same model computed with 3 pressureupdates.


Twophase Flow in 2D

0.15 PVI 0.24 PVI 0.36 PVI

Water flooding in homogeneous domain with highly permeablefractures computed with the first-order upwind scheme.


Model 2 of the SPE Comparative Solution Project

∆t=1 day, 40 minutes ∆t=20 days, 2 minutes

0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Watercut curves computed with the first-order upwind scheme (solidline) and with FrontSim (dashed).


Highlights

Why is this a good ideaExtremely fast solvers: O(n) operations for n unknowns.Local control over Newton iteration.Small memory requirements.Based on well-known conservative discretisation.

Similar to streamline methods in performance: millions of gridcells on desktop computers!


Documents

How to Make Implicit Upwind Schemes as Efficient as Streamline … · 2014. 11. 17. · How to Make Implicit Upwind Schemes as Efﬁcient as Streamline Methods J. R. Natvig†, K.-A